Convergence group

In mathematics, a convergence group or a discrete convergence group is a group acting by homeomorphisms on a compact metrizable space in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary of the hyperbolic 3-space . The notion of a convergence group was introduced by Gehring and Martin (1987) [1] and has since found wide applications in geometric topology, quasiconformal analysis, and geometric group theory.

Formal definition

Let be a group acting by homeomorphisms on a compact metrizable space . This action is called a convergence action or a discrete convergence action (and then is called a convergence group or a discrete convergence group for this action) if for every infinite distinct sequence of elements there exist a subsequence and points such that the maps converge uniformly on compact subsets to the constant map sending to . Here converging uniformly on compact subsets means that for every open neighborhood of in and every compact there exists an index such that for every . Note that the "poles" associated with the subsequence are not required to be distinct.

Reformulation in terms of the action on distinct triples

The above definition of convergence group admits a useful equivalent reformulation in terms of the action of on the "space of distinct triples" of . For a set denote , where . The set is called the "space of distinct triples" for .

Then the following equivalence is known to hold:[2]

Let be a group acting by homeomorphisms on a compact metrizable space with at least two points. Then this action is a discrete convergence action if and only if the induced action of on is properly discontinuous.

Examples

  • The action of a Kleinian group on by Möbius transformations is a convergence group action.
  • The action of a word-hyperbolic group by translations on its ideal boundary is a convergence group action.
  • The action of a relatively hyperbolic group by translations on its Bowditch boundary is a convergence group action.
  • Let be a proper geodesic Gromov-hyperbolic metric space and let be a group acting properly discontinuously by isometries on . Then the corresponding boundary action of on is a discrete convergence action (Lemma 2.11 of [2]).

Classification of elements in convergence groups

Let be a group acting by homeomorphisms on a compact metrizable space with at least three points, and let . Then it is known (Lemma 3.1 in [2] or Lemma 6.2 in [3]) that exactly one of the following occurs:

(1) The element has finite order in ; in this case is called elliptic.

(2) The element has infinite order in and the fixed set is a single point; in this case is called parabolic.

(3) The element has infinite order in and the fixed set consists of two distinct points; in this case is called loxodromic.

Moreover, for every the elements and have the same type. Also in cases (2) and (3) (where ) and the group acts properly discontinuously on . Additionally, if is loxodromic, then acts properly discontinuously and cocompactly on .

If is parabolic with a fixed point then for every one has If is loxodromic, then can be written as so that for every one has and for every one has , and these convergences are uniform on compact subsets of .

Uniform convergence groups

A discrete convergence action of a group on a compact metrizable space is called uniform (in which case is called a uniform convergence group) if the action of on is co-compact. Thus is a uniform convergence group if and only if its action on is both properly discontinuous and co-compact.

Conical limit points

Let act on a compact metrizable space as a discrete convergence group. A point is called a conical limit point (sometimes also called a radial limit point or a point of approximation) if there exist an infinite sequence of distinct elements and distinct points such that and for every one has .

An important result of Tukia,[4] also independently obtained by Bowditch,[2][5] states:

A discrete convergence group action of a group on a compact metrizable space is uniform if and only if every non-isolated point of is a conical limit point.

Word-hyperbolic groups and their boundaries

It was already observed by Gromov[6] that the natural action by translations of a word-hyperbolic group on its boundary is a uniform convergence action (see[2] for a formal proof). Bowditch[5] proved an important converse, thus obtaining a topological characterization of word-hyperbolic groups:

Theorem. Let act as a discrete uniform convergence group on a compact metrizable space with no isolated points. Then the group is word-hyperbolic and there exists a -equivariant homeomorphism .

Convergence actions on the circle

An isometric action of a group on the hyperbolic plane is called geometric if this action is properly discontinuous and cocompact. Every geometric action of on induces a uniform convergence action of on . An important result of Tukia (1986),[7] Gabai (1992),[8] Casson–Jungreis (1994),[9] and Freden (1995)[10] shows that the converse also holds:

Theorem. If is a group acting as a discrete uniform convergence group on then this action is topologically conjugate to an action induced by a geometric action of on by isometries.

Note that whenever acts geometrically on , the group is virtually a hyperbolic surface group, that is, contains a finite index subgroup isomorphic to the fundamental group of a closed hyperbolic surface.

Convergence actions on the 2-sphere

One of the equivalent reformulations of Cannon's conjecture, originally posed by James W. Cannon in terms of word-hyperbolic groups with boundaries homeomorphic to ,[11] says that if is a group acting as a discrete uniform convergence group on then this action is topologically conjugate to an action induced by a geometric action of on by isometries. This conjecture still remains open.

Applications and further generalizations

  • Yaman gave a characterization of relatively hyperbolic groups in terms of convergence actions,[12] generalizing Bowditch's characterization of word-hyperbolic groups as uniform convergence groups.
  • One can consider more general versions of group actions with "convergence property" without the discreteness assumption.[13]
  • The most general version of the notion of Cannon–Thurston map, originally defined in the context of Kleinian and word-hyperbolic groups, can be defined and studied in the context of setting of convergence groups.[14]

References

  1. F. W. Gehring and G. J. Martin, Discrete quasiconformal groups I, Proceedings of the London Mathematical Society 55 (1987), 331–358
  2. B. H. Bowditch, Convergence groups and configuration spaces. Geometric group theory down under (Canberra, 1996), 23–54, de Gruyter, Berlin, 1999.
  3. B. H. Bowditch, Treelike structures arising from continua and convergence groups. Memoirs of the American Mathematical Society 139 (1999), no. 662.
  4. P. Tukia, Conical limit points and uniform convergence groups. Journal für die Reine und Angewandte Mathematik 501 (1998), 71–98
  5. B. Bowditch, A topological characterisation of hyperbolic groups. Journal of the American Mathematical Society 11 (1998), no. 3, 643–667
  6. Gromov, Mikhail (1987). "Hyperbolic groups". In Gersten, Steve M. (ed.). Essays in group theory. Mathematical Sciences Research Institute Publications. 8. New York: Springer. pp. 75–263. doi:10.1007/978-1-4613-9586-7_3. ISBN 0-387-96618-8. MR 0919829.CS1 maint: ref=harv (link)
  7. P. Tukia, On quasiconformal groups. Journal d'Analyse Mathématique 46 (1986), 318–346.
  8. D. Gabai, Convergence groups are Fuchsian groups. Annals of Mathematics 136 (1992), no. 3, 447–510.
  9. A. Casson, D. Jungreis, Convergence groups and Seifert fibered 3-manifolds. Inventiones Mathematicae 118 (1994), no. 3, 441–456.
  10. E. Freden, Negatively curved groups have the convergence property. I. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 20 (1995), no. 2, 333–348.
  11. James W. Cannon, The theory of negatively curved spaces and groups. Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), 315–369, Oxford Sci. Publ., Oxford Univ. Press, New York, 1991
  12. A. Yaman, A topological characterisation of relatively hyperbolic groups. Journal für die Reine und Angewandte Mathematik 566 (2004), 41–89
  13. V. Gerasimov, Expansive convergence groups are relatively hyperbolic, Geometric and Functional Analysis (GAFA) 19 (2009), no. 1, 137–169
  14. W.Jeon, I. Kapovich, C. Leininger, K. Ohshika, Conical limit points and the Cannon-Thurston map. Conformal Geometry and Dynamics 20 (2016), 58–80
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